
where \(Q_{c}\)  is a function of both \(c\)  and \(c_{s}\) . The result given
by Eq. (REF ) explicitly shows that \(c_s<1\)  can
compensate the result of a growing scalar spectrum amplitude the
larger is \(Q\)  and whenever \(c >0\)  (or, likewise, a decreasing
amplitude when \(c<0\) ). Despite the approximations assumed in
Ref. [1]} do allow us to solve for the inflaton
perturbations and, hence, they help to find an analytical expression
for \(\Delta _{\mathcal {R}}\) , it has been shown in
Ref. [2]} that these approximations tend to
overestimate the effect of the coupling between the inflaton and
radiation perturbations. By accounting for the full expressions, i.e.,
not dropping all slow-roll order terms in the equations, the
dependence of \(\Delta _{\mathcal {R}}\)  on \(Q_*\)  tends to be more
suppressed, with a smaller power in \(Q_*\) . Hence, the results from the
(numerical) solution for the full perturbation equations shows that
\(\Delta _{\mathcal {R}} \propto Q_*^\beta \) , with \(\beta < 3 c\) , when
\(c_s=1\) . This result is also observed in our analysis shown in the
next section, showing that \(\beta < 3 c c_{s*}^2\)  also holds here in
the DBIWI. However, the effect is still large enough to make the
spectrum to departure considerably from the result given by
Eq. (REF ). By explicitly numerically solving the set of
perturbations equations, as in Ref. [2]}, the
result of the coupling between inflaton and radiation perturbations
can be expressed as an overall correction to Eq. (REF ), which
modifies it to
\(\Delta _{{\cal R}} = \left(\frac{ H_{*}^2}{2 \pi \dot{\phi }_*}\right)^2\left(1+2n_{ BE} + \frac{2\sqrt{3}\pi Q_*}{\sqrt{3+4\pi Q_*}}{T_*\over H_*}\right) G(Q_*),\) 
